Counting irreducible factors of polynomials over a finite field

✍ Scribed by Arnold Knopfmacher; John Knopfmacher

• Publisher: Elsevier Science
• Year: 1993
• Tongue: English
• Weight: 728 KB
• Volume: 112
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(93)90227-k

✦ Synopsis

Counting irreducible factors of polynomials over a finite field, Discrete Mathematics, 112 (1993) 103-l 18. Let F,[X] denote a polynomial ring in an indeterminate X over a finite field IF,. Exact formulae are derived for (i) the number of polynomials of degree n in F,[X] with a specified number of irreducible factors of a fixed degree r in F,[X]and (ii) the averaye number of such irreducible factors and corresponding oariance for a polynomial of degree n in [F, [ X]. The main emphasis is on the case when multiplicity of factors is counted. These results are then applied to derive the mean and variance for the total number of irreducible factors of polynomials of degree n in F,[X]. n in ff,[X]. Since questions of this nature have been at least partly considered previously for the simpler case of distinct factors (see reference after Theorem 4 below),